This invention relates to a novel titanium trichloride catalytic component which is highly active on polymerization of .alpha.-olefin and a method for advantageously carrying out homopolymerization or copolymerization of .alpha.-olefin in the presence of a catalyst system which consists of the catalytic component and an organoaluminum compound, and more particularly to a titanium trichloride catalytic component manufactured by pulverizing an eutectic mixture or a crystalline compound consisting of titanium trichloride and aluminum chloride (hereinafter will be called an eutectic mixture) or a pulverized matter obtained therefrom (hereinafter will be called a titanium trichloride composite in general) together with an organic acid or an acid anhydride thereof in the absence of an inert solvent at a temperature between 35.degree. and 100.degree. C. throughout the pulverization process, without carrying out any extractive process with such an inert solvent, and, if necessary, by subjecting the matter obtained through such co-pulverization to heat treatment under reduced pressure and a method for carrying out homo-polymerization or copolymerization of .alpha.-olefin in the presence of a catalyst system which consists of the above mentioned titanium trichloride catalytic component and an organic aluminum compound.